Stochastic calculus - Citizendia

Stochastic calculus is a branch of mathematics that operates on stochastic processes. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Albert Einstein and other physical diffusion processes in space of particles subject to random forces. In Mathematics, the Wiener process is a continuous-time Stochastic process named in honor of Norbert Wiener. Norbert Wiener ( November 26, 1894, Columbia Missouri – March 18, 1964, Stockholm, Sweden) was an American This article is about the physical phenomenon for the stochastic process see Wiener process. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Diffusion is the net movement of particles (typically molecules from an area of high concentration to an area of low concentration by uncoordinated random movement Since the 1970's, the Wiener process has been widely applied in financial mathematics to model the evolution in time of stock and bond prices. Mathematical finance is the branch of Applied mathematics concerned with the Financial markets.

The main flavours of stochastic calculus are the Itō calculus and its variational relative the Malliavin calculus. Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process) The Malliavin calculus, named after Paul Malliavin, is a theory of variational Stochastic calculus. For technical reasons the Itō integral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines. In Stochastic processes the Stratonovich integral (developed simultaneously by Ruslan L ) The Stratonovich integral can readily be expressed in terms of the Itō integral. Another benefit of the Stratonovich integral is that it enables some problems to be expressed in a co-ordinate system invariant form and is therefore invaluable when developing stochastic calculus on manifolds other than Rⁿ. The Dominated convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itō form. In Measure theory, a branch of Mathematical analysis, Lebesgue 's dominated convergence theorem provides Sufficient conditions under which two

Itō integral

Main article: Itō calculus

The Itō integral is central to the study of stochastic calculus. Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process) Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to Stochastic processes such as Brownian motion ( Wiener process) The integral $\int H\,dX$ is defined for a semimartingale X and locally bounded predictable process H. In probability theory a real valued process X is called a semimartingale if it can be decomposed as the sum of a Local martingale and an adapted finite-variation

Stratonovich integral

Main article: Stratonovich integral

The Stratonovich integral can be defined in terms of the Itō integral as

$\int_0^t X_s \circ d Y_s : = \int_0^t X_s d Y_s + \frac{1}{2} \left [ X, Y\right]_t.$

The alternative notation

$\int_0^t X_s \partial Y_s$

is also used to denote the Stratonovich integral. In Stochastic processes the Stratonovich integral (developed simultaneously by Ruslan L

External links

Notes on Stochastic Calculus — A short elementary description of the basic Itō integral.

T. Szabados and B. Szekely, Stochastic integration based on simple, symmetric random walks - A new approach which the authors hope is more transparent and technically less demanding.

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